Ensuring All Students Learn Math 6 – 12 - Schedschd.ws/hosted_files/losdpd2016/e3/Schuhl Handouts...
Transcript of Ensuring All Students Learn Math 6 – 12 - Schedschd.ws/hosted_files/losdpd2016/e3/Schuhl Handouts...
EnsuringAllStudentsLearnMath6–12
Twi=er:@SSchuhl
SessionLearningTargets
§ Wecaniden)fytheessen)alcontentstandardsstudentsmustlearninmathema)cs.
§ Wecaniden)fyhighcogni)vetaskstouseasforma)vefeedbackduringinstruc)on.
§ Wecandesignwaystoremediateandintervenewhenstudentsstruggletolearnmathema)cs.
Star)ng…Ge=ngThere…GotIt!
Star)ng…Ge=ngThere…GotIt!
Star)ng…Ge=ngThere…GotIt!
WhatDoWeExpect StudentsToLearn?
1
SnapshotofCognitiveRigorMatrix:MathematicsThismatrixfromtheSmarterBalancedGeneralItemSpecificationsdrawsfrombothBloom’s(revised)TaxonomyofEducationalObjectivesandWebb’sdepth-of-knowledgelevels. DepthofThinking
(Webb)+TypeofThinking(RevisedBloom,
2001)
DOKLevel-1RecallandReproduction
DOKLevel-2BasicSkills&Concepts
DOKLevel-3StrategicThinkingandReasoning
DOKLevel-4ExtendedThinking
Remember • Recallconversions,terms,facts.
Understand
• Evaluateanexpression.
• Locatepointsonagridornumberonnumberline.
• Solveaone-stepproblem.
• Representmathrelationshipsinwords,pictures,orsymbols.
• Specify,explainrelationships
• Makebasicinferencesorlogicalpredictionsfromdataandobservations.
• Usemodelsanddiagramstoexplainconcepts.
• Makeandexplainestimates.
• Useconceptstosolvenonroutineproblems.Usesupportingevidencetojustifyconjectures,generalize,orconnectideas.
• Explainreasoningwhenmorethanoneresponseispossible.
• Explainphenomenaintermsofconcepts.
• Relatemathematicalconceptstoothercontentareas,otherdomains.
• Developgeneralizationsoftheresultsobtainedandthestrategiesusedandapplythemtonewproblemsituation.
Apply
• Followsimpleprocedures.
• Calculate,measure,applyarule(e.g.,rounding).
• Applyalgorithmorformula.
• Solvelinearequations.• Makeconversions.
• Selectaprocedureandperformit.
• Solveroutineproblemapplyingmultipleconceptsordecisionpoints.
• Retrieveinformationtosolveaproblem.
• Translatebetweenrepresentations.
• Designinvestigationforaspecificpurposeorresearchquestion.
• Usereasoning,planning,andsupportingevidence.
• Translatebetweenproblemandsymbolicnotationwhennotadirecttranslation.
• Initiate,design,andconductaprojectthatspecifiesaproblem,identifiessolutionpaths,solvestheproblem,andreportsresults.
Analyze
• Retrieveinformationfromatableorgraphtoansweraquestion.
• Identifyapatternortrend.
• Categorizedata,figures.
• Organize,orderdata.• Selectappropriategraphandorganizeanddisplaydata.
• Interpretdatafromasimplegraph.
• Extendapattern.
• Compareinformationwithinoracrossdatasetsortexts.
• Analyzeanddrawconclusionsfromdata,citingevidence.
• Generalizeapattern.• Interpretdatafrom
complexgraph.
• Analyzemultiplesourcesofevidenceordatasets.
Evaluate
• Citeevidenceanddevelopalogicalargument.
• Compareandcontrastsolutionmethods.
• Verifyreasonableness.
• Applyunderstandinginanovelway.
• Provideargumentorjustificationforthenewapplication.
Create
• Brainstormideas,concepts,problems,orperspectivesrelatedtoatopicorconcept.
• Generateconjecturesorhypothesesbasedonobservationsorpriorknowledgeandexperience.
• Developanalternativesolution.
• Synthesizeinformationwithinonedataset.
• Synthesizeinformationacrossmultiplesourcesordatasets.
• Designamodeltoinformandsolveapracticalorabstractsituation.
2
AWordAboutRigor
AcademicandInstruc)onalRigor
RigoronAssessmentDOK
StandardsforMathemaHcalPracHce
1. Make sense of problems and persevere in solving them.
6. Attend to precision.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
AccessandEquity“Anexcellentmathema)csprogramrequiresthatallstudentshaveaccesstoahighqualitymathema)cscurriculum,effec)veteachingandlearning,highexpecta)ons,andthesupportandresourcesneededtomaximizetheirlearningpoten)al.”
—NCTM,PrinciplestoAc.ons,p.59
3
WhatDoWeWantStudentstoLearn?
AGuaranteedandViableCurriculum
§ Intended:whatwewantthemtolearn
§ Implemented:whatactuallygetstaught
§ A=ained:whattheyactuallylearn
(Marzano,WhatWorksinSchools:Transla.ngResearchIntoAc.on,2003)
“Tobeginwiththeendinmindmeanstostartwithaclearunderstandingofyourdes)na)on.
“Itmeanstoknowwhereyou’regoingsothatyoubeWerunderstandwhereyouarenowsothatthestepsyoutakearealwaysintherightdirec)on.”
—Covey,TheSevenHabitsofHighlyEffec.vePeople:PowerfulLessonsin
PersonalChange(1994)
Standards§ Whichstandardsarepriority?
o Allstudentswillbegivenaddi)onal)meandsupport,ifneeded
o Focusofinstruc)onalandassessment)meandenergy
§ WhichstandardsaresupporHng?
4
WhatArePriorityStandards?§ Asubsetoftheen)relistofstatecontentandperformancestandards
§ Prioritystandardsessen)alforstudentunderstandingandsuccess
§ Thestandardseachteacherneedstoensureeverystudentlearnspriortoleavingthecurrentgrade
§ Abrief,straigh[orward,easytoreaddocumenttoguidestandards-basedinstruc)on
Essen)alstandards“donotrelieveteachersoftheresponsibilityforteachingallstandardsandindicators,butdoiden)fywhichstandardsarecriHcalforstudentsuccessandwhichonescanbegivenlessemphasis.”
BigIdea
—Ainsworth,PowerStandards:Iden.fyingtheStandardsThatMaGerMost(2003),p.11
(p.5)
IstheStandardEssen-al?§ Doesithaveendurance?§ Doesithaveleverage?§ Doesitdevelopstudentreadinessforthe
nextleveloflearning?
(Reeves,TheLeader’sGuidetoStandards:ABlueprintfor
Educa.onalEquityandExcellence,2002)
5
CC
SSM
Pri
ori
ty C
lust
ers
6–1
1
Gra
de
6 G
rad
e 7
Gra
de
8 G
rad
e 11
R
atio
s an
d P
rop
ort
ion
al
Re
aso
nin
g U
nd
erst
an
d r
ati
o c
on
cep
ts a
nd
u
se r
ati
o r
easo
nin
g to
so
lve
pro
ble
ms.
The
Nu
mb
er
Syst
em
A
pp
ly a
nd
ext
end
pre
vio
us
un
der
sta
ndi
ngs
of
mu
ltip
lica
tio
n a
nd
div
isio
n t
o
div
ide
fra
ctio
ns b
y fr
acti
ons
.
Ap
ply
an
d e
xten
d p
revi
ou
s u
nd
erst
an
din
gs o
f n
um
ber
s to
the
syst
em o
f ra
tio
nal
n
um
ber
s.
Exp
ress
ion
s an
d E
qu
atio
ns
Ap
ply
an
d e
xten
d p
revi
ou
s u
nd
erst
an
din
gs o
f ar
ith
met
ic t
o
alge
bra
ic e
xpre
ssio
ns.
Rea
son
ab
ou
t a
nd
sol
ve o
ne-
vari
abl
e eq
ua
tio
ns a
nd
in
equ
alit
ies.
Rep
rese
nt
an
d a
nal
yze
qu
an
tita
tive
rel
ati
on
ship
s b
etw
een
dep
end
ent
an
d
ind
epen
den
t va
ria
bles
.
Rat
ios
and
Pro
po
rtio
nal
R
eas
on
ing
An
aly
ze p
rop
ort
iona
l re
lati
on
ship
s a
nd
use
th
em
to
solv
e re
al-
wo
rld
an
d
ma
the
ma
tica
l p
robl
ems.
The
Nu
mb
er
Syst
em
A
pp
ly a
nd
ext
end
pre
vio
us
un
der
sta
ndi
ngs
of
op
era
tio
ns
wit
h f
ract
ion
s to
ad
d, s
ub
trac
t,
mu
ltip
ly, a
nd
div
ide
rati
ona
l n
um
ber
s.
Exp
ress
ion
s an
d E
qu
atio
ns
Use
pro
per
ties
of
op
era
tio
ns t
o
gen
era
te e
qu
iva
len
t ex
pre
ssio
ns.
Solv
e re
al-
life
an
d
ma
the
ma
tica
l p
robl
ems
usin
g
nu
mer
ical
an
d a
lgeb
raic
ex
pre
ssio
ns a
nd
eq
ua
tio
ns.
Exp
ress
ion
s an
d E
qu
atio
ns
Wo
rk w
ith
ra
dica
ls a
nd
inte
ger
exp
on
ents
.
Un
der
sta
nd
th
e co
nn
ecti
on
s b
etw
een
pro
po
rtio
nal
rela
tio
nsh
ips,
lin
es, a
nd
lin
ear
equ
ati
on
s.
An
aly
ze a
nd
sol
ve li
nea
r
equ
ati
on
s a
nd
pai
rs o
f si
mu
lta
neo
us
lin
ear
equ
ati
ons
.
Fun
ctio
ns
Def
ine,
eva
lua
te, a
nd
co
mp
are
fun
ctio
ns.
Ge
om
etr
y U
nd
erst
an
d c
on
gru
ence
an
d
sim
ilari
ty u
sin
g p
hys
ical
mo
del
s,
tra
nsp
are
nci
es, o
r ge
om
etr
y so
ftw
are
.
Un
der
sta
nd
an
d a
ppl
y th
e P
yth
ago
rea
n T
heo
rem
.
Seei
ng
the
Str
uct
ure
in E
xpre
ssio
ns
Inte
rpre
t th
e st
ruct
ure
of
exp
ress
ions
.
Wri
te e
xpre
ssio
ns in
eq
uiv
alen
t fo
rms
to s
olv
e p
rob
lem
s.
Ari
thm
eti
c w
ith
Pol
yno
mia
ls a
nd
Rat
ion
al
Exp
ress
ion
s P
erfo
rm a
rith
met
ic o
per
ati
ons
on
pol
yno
mia
ls.
Cre
atin
g Eq
uat
ion
s
Cre
ate
eq
ua
tio
ns
tha
t d
escr
ibe
nu
mb
ers
or
rela
tio
nshi
ps.
Re
aso
nin
g w
ith
Eq
uat
ion
s an
d In
eq
ual
itie
s U
nd
erst
an
d s
olvi
ng
equ
ati
on
s as
a p
roce
ss o
f re
aso
nin
g a
nd
exp
lain
th
e re
aso
nin
g.
Solv
e eq
ua
tio
ns
an
d in
equ
alit
ies
in o
ne
vari
abl
e.
Rep
rese
nt
an
d s
olve
eq
ua
tio
ns
an
d in
equ
alit
ies
gra
phi
call
y.
Inte
rpre
ting
Fu
nct
ion
s
Un
der
sta
nd
th
e co
nce
pt
of
a f
un
ctio
n a
nd
un
der
sta
nd
fu
nct
ion
no
tati
on
.
Inte
rpre
t fu
nct
ion
s th
at
ari
se i
n a
ppl
ica
tio
ns in
ter
ms
of
the
con
text
.
An
aly
ze f
un
ctio
ns
usin
g di
ffer
ent
rep
rese
nta
tio
ns.
Bu
ild
ing
Fun
ctio
ns
B
uil
d a
fu
nct
ion
th
at
mo
del
s a
rel
ati
ons
hip
bet
we
en t
wo
qu
an
titi
es.
Sour
ce: T
able
com
pile
d vi
a in
form
atio
n ac
cess
ed a
t the
follo
win
g U
RLs:
www.sm
arterb
alan
ced.or
g/wp-
cont
ent/u
ploa
ds/2
015/
08/M
athe
matics_
Blue
prin
t.pdf
ht
tp://
achi
evethe
core
.org
/con
tent
/upl
oad/
Focu
s_in
_Math_
06.12.20
13.pdf
6
Priority Standards—Course or Grade Level: __________________________ Identify the essential standards for each course and the reasons each is priority.
Priority Standards
End
uran
ce
Lev
erag
e
Rea
dine
ss
7
PriorityStandards:VerticalConnections
1.Readyourstandards.Whichonesarepriority(haveendurance,leverage,andprovidereadinessforthenextcourse)?2.Whatare7–10essentialskillsstudentsinmygradeorcoursemustlearn?3.Whatare7–10essentialskillsstudentsshouldcometomygradeorcoursehavinglearned?
8
R E P R O D U C I B L E72 |
Simplifying Response to Intervention © 2012 Solution Tree Press • solution-tree.com Visit go.solution-tree.com/rti to download this page.
Essential Standards ChartW
hat
Is It
We
Exp
ect
Stud
ents
to
Lea
rn?
Gra
de:
Sub
ject
:S
emes
ter:
Team
Mem
ber
s:
Des
crip
tio
n o
f St
and
ard
E
xam
ple
of
Rig
or
Pre
req
uisi
te
Skill
s W
hen
Taug
ht?
Co
mm
on
Sum
mat
ive
Ass
essm
ent
Ext
ensi
on
Stan
dar
ds
Wha
t is
the
es
sent
ial s
tand
ard
to
be
lear
ned
? D
escr
ibe
in
stud
ent-
frie
ndly
vo
cab
ular
y.
Wha
t d
oes
p
rofi
cien
t st
uden
t w
ork
loo
k lik
e?
Pro
vid
e an
ex
amp
le a
nd/o
r d
escr
ipti
on.
Wha
t p
rio
r
kno
wle
dg
e, s
kills
, an
d/o
r vo
cab
ular
y ar
e ne
eded
fo
r a
stud
ent
to m
aste
r th
is s
tand
ard
?
Whe
n w
ill t
his
stan
dar
d b
e ta
ught
?
Wha
t as
sess
men
t(s)
w
ill b
e us
ed to
m
easu
re s
tud
ent
mas
tery
?
Wha
t w
ill w
e d
o
whe
n st
uden
ts
have
alr
ead
y le
arne
d t
his
stan
dar
d?
page 1 of 2
9
R E P R O D U C I B L E | 73
Simplifying Response to Intervention © 2012 Solution Tree Press • solution-tree.com Visit go.solution-tree.com/rti to download this page.
page 2 of 2
Wo
rkin
g in
co
llab
ora
tive
tea
ms,
exa
min
e al
l rel
evan
t d
ocu
men
ts, c
om
mo
n c
ore
sta
nd
ard
s, s
tate
sta
nd
ard
s, a
nd
d
istr
ict
po
wer
sta
nd
ard
s, a
nd
th
en a
pp
ly t
he
crit
eria
of
end
ura
nce
, lev
erag
e, a
nd
rea
din
ess
to d
eter
min
e w
hic
h st
and
ard
s ar
e es
sen
tial
fo
r al
l stu
den
ts t
o m
aste
r. R
emem
ber
, les
s is
mo
re. F
or
each
sta
nd
ard
sel
ecte
d, c
om
ple
te
the
rem
aini
ng c
olu
mns
. Co
mp
lete
thi
s ch
art
by
the
seco
nd o
r th
ird
wee
k o
f ea
ch in
stru
ctio
nal p
erio
d (
sem
este
r).
10
2007
–200
8 Se
cond
Sem
este
r Ess
entia
l Sta
ndar
ds
Cou
rse
Title
: Alg
ebra
1
Team
Mem
bers
: Jac
kie
Mar
tin, B
re W
elch
, Jac
kie
Stoe
rger
, Mar
y H
ings
t
Stan
dard
St
anda
rd o
r Des
crip
tion
Exam
ple
and
Rig
or
Prio
r Ski
lls N
eede
d C
omm
on
Ass
essm
ent
Whe
n Ta
ught
2.0
10.0
Stu
dent
s un
ders
tand
and
use
the
rule
s of
ex
pone
nts.
Stu
dent
s m
ultip
ly a
nd d
ivid
e m
onom
ials
.
Sim
plify
: 3
7 9
5 10xy xy
Mul
t iply
ing
mon
omia
ls a
nd
poly
nom
ials
(Cha
pter
4)
Cha
pter
4 C
A
Feb.
11.0
Stu
dent
s ap
ply
basi
c fa
ctor
ing
tech
niqu
es to
se
cond
- and
sim
ple
third
-deg
ree
poly
nom
ials
. Th
ese
tech
niqu
es in
clud
e fin
ding
a c
omm
on
fact
or fo
r all
term
s in
a p
olyn
omia
l, re
cogn
izin
g th
e di
ffere
nce
of tw
o sq
uare
s, a
nd
reco
gniz
ing
perfe
ct s
quar
es o
f bin
omia
ls.
Fact
or c
ompl
etel
y:
1.3a
2 –
24ab
+ 4
8b2
2.x2 –
121
3.9x
2 + 1
2x +
4
Mul
tiply
ing
and
divi
ding
m
onom
ials
and
po
lyno
mia
ls (C
hapt
er 4
and
C
hapt
er 5
: Sec
. 1–3
)
Cha
pter
5 C
A
Feb.
12.0
S
t ude
nts
sim
plify
frac
tions
with
pol
ynom
ials
in
the
num
erat
or a
nd d
enom
inat
or b
y fa
ctor
ing
both
and
redu
cing
them
to th
e lo
wes
t ter
ms.
Sim
plify
: 316
8+
22
2
44
36
xxy
yxy
y!
+
!
Fact
orin
g by
find
ing
GC
F,
diffe
renc
e of
two
squa
res,
an
d tri
nom
ials
(Cha
pter
5)
Cha
pter
6 C
A
Mar
ch
2.0
St u
dent
s un
ders
tand
and
use
the
oper
atio
n of
ta
king
a ro
ot a
nd ra
isin
g to
a fr
actio
nal p
ower
. S
impl
ify: 3
168
+
Und
erst
andi
ng ra
tiona
l and
irr
atio
nal n
umbe
rs a
nd
prim
e fa
ctor
ing
Cha
pter
11:
S
ec. 3
, 4, 5
C
A
Mar
ch
14.0
S
olve
a q
uadr
atic
equ
atio
n by
fact
orin
g or
co
mpl
etin
g th
e sq
uare
. S
olve
by
com
plet
ing
the
squa
re:
x2 +
4x =
6
Fact
orin
g qu
adra
tics
(Cha
pter
5) a
nd s
impl
ifyin
g ra
dica
ls (C
hapt
er 1
1)
Cha
pter
12:
S
ec. 1
–4 a
nd
Cha
pter
5:
Sec
. 12
CA
Late
M
arch
21.0
S
tude
nts
grap
h qu
adra
tic fu
nctio
ns a
nd k
now
th
at th
eir r
oots
are
the x-
inte
rcep
ts.
Gra
ph:
y =
x2 – 3
x –
4 an
d st
ate
the
x in
terc
epts
.
Sol
ving
qua
drat
ic e
quat
ions
by
fact
orin
g, c
ompl
etin
g th
e sq
uare
, and
qua
drat
ic
form
ula
(Cha
pter
12)
Cha
pter
8:
Sec
. 8 a
nd
p.38
9 C
AA
pril
REPRODUCIBLE
© Buffum, Mattos, & Weber 2012. solution-tree.comReproducible.
REPRODUCIBLE
328RTI at Work Workshop
© Solution Tree 2014 • solution-tree.com • Reproducible.
REPRODUCIBLE
11
Esse
ntia
l Sta
ndar
ds S
tude
nt T
rack
ing
Cha
rt
Esse
ntia
l Sta
ndar
d C
omm
on A
sses
smen
t D
ate
Pass
edTe
ache
rIn
itial
s
RTI at Work Workshop© Solution Tree 2014 • solution-tree.com • Reproducible.
REPRODUCIBLE
12
PlanningFromStandardsandAssessment
Standard
TargetTargetTargetTarget
Assessment
Curriculum
Instruc)on
(Wiggins&McTighe,UnderstandingbyDesign,2000)
UnitPlanning
CFA CFA CSA20daysforunit–2daysforcommonunitassessment(CSA)andreview–2daysforCFAandresponse–2bufferdaysforre-engagementinlearning=14daysforTier1coreinstruc)on
Howwillyoumaximizelearningeachofthe14days?
Algebra1ProficiencyMap
13
ProficiencyMapChecklistAproficiencymapidentifiesstandardsstudentsshouldmasterybytheendoftheunit.Unittitlesarelistedacrossthetopofthechart,includingthenumberofdaysallocatedforteaching.Domainsorstrandsarelistedalongtheleftcolumn.Teachersinsertthestandards.Sometimesastandardmayneedtobelistedinmorethanoneunit.Ifso,teachersshouldidentifypartsofthestandardforstudentstomaster.Teachersplaceanasterisknexttostandardstoidentifywhentheyarelistedinmorethanoneunit.Underlineprioritystandards.ExampleProficiencyMapChecklist
1. Iseverystandardlistedonetimewhenproficiencyisexpected?Ifpartofastandardislistedinoneunit,istherestofthestandardaccountedforanddobothpartshaveanasterisk?
2. Aretheprioritystandardsidentifiedineachunit?
3. Doeseveryunithaveanameandanumberofdays?Are155daysaccountedforonthetotalprojectionmap?(Daysmarkedbyanasteriskincludeassessments.)
4. Howishorizontalcoherencebuiltintotheproficiencymap?Forexample,howcanprevious
conceptsfromtheyearbewoveninorsupportlearninginalaterunit?
5. Howisverticalcoherencebuiltintotheproficiencymap?Forexample,whatdidstudentslearnlastyearandwhen?Whatwilltheyusethislearningfornextyearandwhen?Foritemsmarkedbyanasterisk,youmayneedtolookattheproficiencymapsforthegradelevelaboveandbelowyourown.
Grade 5
Math
Multiplication & Division 25 days
Volume of Rectangular
Prisms 15 days
Decimals & Conversions 35 days
Fractions: Addition & Subtraction
25 days
Fractions: Division &
Multiplication 35 days
Graphing & Geometry
15 days
OA
*5.OA.1 Evaluate expressions with parenthesis (whole numbers) *5.OA.2 Write and interpret expressions (whole numbers).
*5.OA.1 Evaluate expressions with parenthesis (with powers of ten).
*5.OA.1 Evaluate expressions with parenthesis (fractions).
5.OA.3 Know number patterns.
NBT
5.NBT.5 Use standard algorithm for multiplication. 5.NBT.6 Use models for division.
5.NBT.1 Know place value with tens. 5.NBT.2 ×÷ by 10 5.NBT.3a Read and write decimals. 5.NBT.3b Compare decimals. 5.NBT.4 Round decimals. 5.NBT.7 Add, subtract, multiply, and divide decimals.
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SessionLearningTargets
§ Wecaniden)fytheessen)alcontentstandardsstudentsmustlearninmathema)cs.
§ Wecaniden)fyhighcogni)vetaskstouseasforma)vefeedbackduringinstruc)on.
§ Wecandesignwaystoremediateandintervenewhenstudentsstruggletolearnmathema)cs.
Star)ng…Ge=ngThere…GotIt!
Star)ng…Ge=ngThere…GotIt!
Star)ng…Ge=ngThere…GotIt!
Unproduc)veBeliefs Produc)ve
Beliefs
“Itisimportanttonotethatthesebeliefsshouldnotbeviewedasgoodorbad.Instead,beliefsshouldbeunderstoodasunproduc)vewhentheyhindertheimplementa)onofeffec)veinstruc)onalprac)ceorlimitstudentaccesstoimportantmathema)cscontentandprac)ces.”
—NCTM,PrinciplestoAc.ons(2014),p.11
Low-LevelandHigh-LevelTasks
Low-LevelTasks§ Memoriza)on§ Procedureswithoutconnec)onstomeaning§ Algorithmic
High-LevelTasks§ Procedureswithconnec)onstomeaning§ Doingmathema)cs§ Nonrou)ne
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(NCTM, page 1 of 2)
NCTM:PrinciplestoActions
(Source:ThefollowingteachingpracticesareexcerptedfromNationalCouncilofTeachersofMathematics,PrinciplestoActions:EnsuringMathematicalSuccessforAll,2014.ThePrinciplestoActionsexecutivesummaryisavailableatwww.nctm.org/uploadedFiles/Standards_and_Positions/PtAExecutiveSummary.pdffordownload.)
MathematicsTeachingPractices
Establishmathematicsgoalstofocuslearning.Effectiveteachingofmathematicsestablishescleargoalsforthemathematicsthatstudentsarelearning,situatesgoalswithinlearningprogressions,andusesthegoalstoguideinstructionaldecisions.
Implementtasksthatpromotereasoningandproblemsolving.Effectiveteachingofmathematicsengagesstudentsinsolvinganddiscussingtasksthatpromotemathematicalreasoningandproblemsolvingandallowmultipleentrypointsandvariedsolutionsstrategies.
Useandconnectmathematicalrepresentations.Effectiveteachingofmathematicsengagesstudentsinmakingconnectionsamongmathematicalrepresentationstodeepenunderstandingofmathematicsconceptsandproceduresandastoolsforproblemsolving.
Facilitatemeaningfulmathematicaldiscourse.Effectiveteachingofmathematicsfacilitatesdiscourseamongstudentstobuildsharedunderstandingofmathematicalideasbyanalyzingandcomparingstudentapproachesandarguments.
Posepurposefulquestions.Effectiveteachingofmathematicsusespurposefulquestionstoassessandadvancestudents'reasoningandsensemakingaboutimportantmathematicalideasandrelationships.
Buildproceduralfluencyfromconceptualunderstanding.Effectiveteachingofmathematicsbuildsfluencywithproceduresonafoundationofconceptualunderstandingsothatstudents,overtime,becomeskillfulinusingproceduresflexiblyastheysolvecontextualandmathematicalproblems.
Supportproductivestruggleinlearningmathematics.Effectiveteachingofmathematicsconsistentlyprovidesstudents,individuallyandcollectively,withopportunitiesandsupportstoengageinproductivestruggleastheygrapplewithmathematicalideasandrelationships.
Elicitanduseevidenceofstudentthinking.Effectiveteachingofmathematicsusesevidenceofstudentthinkingtoassessprogresstowardmathematicalunderstandingandtoadjustinstructioncontinuallyinwaysthatsupportandextendlearning.
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(NCTM, page 2 of 2)
BeliefsAboutTeachingandLearningMathematics
Thefollowingtableshowtheunproductiveandproductivebeliefsofmathematicsteachersthatinfluencetheirteachingandstudents’learningofmathematics(NCTM,p.11).
UnproductiveBeliefs ProductiveBeliefs
Mathematicslearningshouldfocusonpracticingproceduresandmemorizingbasicnumbercombinations.
Mathematicslearningshouldfocusondevelopingunderstandingofconceptsandproceduresthroughproblemsolving,reasoning,anddiscourse.
Studentsneedonlytolearnandusethesamestandardcomputationalalgorithmsandthesameprescribedmethodstosolvealgebraicproblems.
Allstudentsneedtohavearangeofstrategiesandapproachesfromwhichtochooseinsolvingproblems,including,butnotlimitedto,generalmethods,standardalgorithms,andprocedures.
Studentscanlearntoapplymathematicsonlyaftertheyhavemasteredthebasicskills.
Studentscanlearnmathematicsthroughexploringandsolvingcontextualandmathematicalproblems.
Theroleoftheteacheristotellstudentsexactlywhatdefinitions,formulas,andrulestheyshouldknowanddemonstratehowtousethisinformationtosolvemathematicsproblems.
Theroleoftheteacheristoengagestudentsintasksthatpromotereasoningandproblemsolvingandfacilitatediscoursethatmovesstudentstowardsharedunderstandingofmathematics.
Theroleofthestudentistomemorizeinformationthatispresentedandthenuseittosolveroutineproblemsonhomework,quizzes,andtests.
Theroleofthestudentistobeactivelyinvolvedinmakingsenseofmathematicstasksbyusingvariedstrategiesandrepresentations,justifyingsolutions,makingconnectionstopriorknowledgeorfamiliarcontextsandexperiences,andconsideringthereasoningofothers.
Aneffectiveteachermakesthemathematicseasyforstudentsbyguidingthemstepbystepthroughproblemsolvingtoensurethattheyarenotfrustratedorconfused.
Aneffectiveteacherprovidesstudentswithappropriatechallenge,encouragesperseveranceinsolvingproblems,andsupportsproductivestruggleinlearningmathematics.
Theproductivebeliefsmirrorthemathematicsteachingpracticesthatbenefitstudentlearning.Howcanyourteamincorporatetheinstructionalpracticeslistedontheproductivebeliefscolumnofthechart?
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141
APPENDIX C
Cognitive-Demand-Level Task-Analysis GuideSource: Smith & Stein, 1998. © 1998, National Council of Teachers of Mathematics. Used with permission.
Table C.1: Cognitive-Demand Levels of Mathematical Tasks
Lower-Level Cognitive Demand Higher-Level Cognitive Demand
Memorization Tasks
These tasks involve reproducing previously learned facts, rules, formulae, or definitions to memory.
They cannot be solved using procedures because a procedure does not exist or because the time frame in which the task is being completed is too short to use the procedure.
They are not ambiguous; such tasks involve exact reproduction of previously seen material and what is to be reproduced is clearly and directly stated.
They have no connection to the concepts or meaning that underlie the facts, rules, formulae, or definitions being learned or reproduced.
Procedures With Connections Tasks
These procedures focus students’ attention on the use of procedures for the purpose of developing deeper levels of understanding of mathematical concepts and ideas.
They suggest pathways to follow (explicitly or implicitly) that are broad general procedures that have close connections to underlying conceptual ideas as opposed to narrow algorithms that are opaque with respect to underlying concepts.
They usually are represented in multiple ways (for example, visual diagrams, manipulatives, symbols, or problem situations). They require some degree of cognitive effort. Although general procedures may be followed, they cannot be followed mindlessly. Students need to engage with the conceptual ideas that underlie the procedures in order to successfully complete the task and develop understanding.
Procedures Without Connections Tasks
These procedures are algorithmic. Use of the procedure is either specifically called for, or its use is evident based on prior instruction, experience, or placement of the task.
They require limited cognitive demand for successful completion. There is little ambiguity about what needs to be done and how to do it.
They have no connection to the concepts or meaning that underlie the procedure being used.
They are focused on producing correct answers rather than developing mathematical understanding.
They require no explanations or have explanations that focus solely on describing the procedure used.
Doing Mathematics Tasks
Doing mathematics tasks requires complex and no algorithmic thinking (for example, the task, instructions, or examples do not explicitly suggest a predictable, well‐rehearsed approach or pathway).
It requires students to explore and understand the nature of mathematical concepts, processes, or relationships.
It demands self‐monitoring or self‐regulation of one’s own cognitive processes.
It requires students to access relevant knowledge and experiences and make appropriate use of them in working through the task.
It requires students to analyze the task and actively examine task constraints that may limit possible solution strategies and solutions.
It requires considerable cognitive effort and may involve some level of anxiety for the student due to the unpredictable nature of the required solution process.
Beyond the Common Core, Leader’s Guide ©2015SolutionTreePress•SolutionTree.com
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MathTasks(A-SSE.A.1)Completing the square. Use completing the square to solve x2 – 4x – 8 = 0 Profit of a company The profit, P (in thousands of dollars), that a company makes selling an item is a quadratic function of the price, x, (in dollars), that they charge for the item. The following expressions are equivalent:
!!
P(x)= −2x2 +24x −54P(x)= −2(x −3)(x −9)P(x)= −2(x −6)2 +18
1. Which of the equivalent expressions for P(x) reveals the price which gives a profit of zero without changing the form of the expression? Find a price which gives a profit of zero.
2. Which of the equivalent expressions for P(x) reveals the profit when the price is zero without changing the form of the expression? Find the profit when the price is zero.
3. Which of the equivalent expressions for P(x) reveals the price which produces the highest possible profit without changing the form of the expression? Find the price which gives the highest possible profit. ---www.illustrativemathematics.org
List the similarities and differences between the two problems.
Similarities Differences
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CharacterisHcsofTeacherQuesHonsThatSupportStudents’AccesstoContent
AssessingQuesHonsPrepareques)onstoscaffoldinstruc)onforstudentswhoarestuckduringatask.
AdvancingQuesHonsPrepareques)onstofurtherlearningforstudentswhoarereadytoadvancebeyondlearningtargettasks.
FromProblemPerformerstoProblemSolvers
ProblemPerformers Problemsolvers
Doitjustliketheteacherdid. Thinkbeforeyoustart.
Writeyournumbersstraight. Drawapicturetounderstandtheproblem.
Don’ttalkwhenyouwork.That’schea)ng.
Explainhowyousolvedit,usingmathvocabulary.
Findananswerandmoveon. Checkyourwork.Doitadifferentway.
SessionLearningTargets
§ Wecaniden)fytheessen)alcontentstandardsstudentsmustlearninmathema)cs.
§ Wecaniden)fyhighcogni)vetaskstouseasforma)vefeedbackduringinstruc)on.
§ Wecandesignwaystoremediateandintervenewhenstudentsstruggletolearnmathema)cs.
Star)ng…Ge=ngThere…GotIt!
Star)ng…Ge=ngThere…GotIt!
Star)ng…Ge=ngThere…GotIt!
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ConnecHons
6in
4in24in2
20 6
30
4
600 180
80 24
26×34=884
2x 6
x
4
2x2 6x
8x 24
(x+4)(2x+6)=2x2+14x+24
6
4 24 2
3
12
12
14
6 12 × 4 12 = 29 1
4
“Thebestinterven)onispreven)on.”
--MikeMaWos
WhatneedstobeinTier1
CoreInstrucHon?
(p.47)
FormaHveAssessmentProcessesRequireTwoAddiHonalComponents
1. Meaningfulfeedbacktostudents
FeedbackmustbeFAST:
Fair
Accurate
Specific
Timely
2.StudentacHononfeedback
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EffecHveClassroomFormaHveAssessment§ Howdostudentsexpresstheir
ideas,ques)ons,insights,anddifficul)es?
§ Wherearethemostsignificantconversa)onstakingplace(studenttoteacher,studenttostudent,teachertostudent)?
Dostudentsseeeachotherasreliableandvaluable
resources?
3CriHcalInstrucHonQuesHons
§ Howdowestructurestudenttostudentdiscourseforatleast65%ofeachlesson?
§ Howdowemakelearningvisible?
§ Howdostudentsknowiftheylearnedthelearningtargetforthedayornot?
HowInterveneand/orRemediate?§ Chooseastandard.
§ Howmightstudentsstruggle?
§ Forinterven)on,considernonalgorithmicwaystomakesenseoftheconcept.Whatvisualmodelscanbeused?
§ Forremedia)on,considertheprerequisiteskillsneededtomakesenseoftheconcept.Howcanthesebeincludedwithinterven)on?
Whenintheschooldaycanthesebeaddressed?
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R E P R O D U C I B L E | 115
Simplifying Response to Intervention © 2012 Solution Tree Press • Visit go.solution-tree.com/rti to download this page.
Plan for Remediation Plan for Intervention Plan for EnrichmentBased on the prior skills needed, how will we determine which stu-dents need remediation before we begin initial instruction? Who will conduct the remedia-tions? When?
After initial instruction and differentiation, what is our team’s plan to provide additional time and support to those students who have not learned? Who will con-duct the interventions? When?
After initial instruction and intervention, what is our team’s plan to provide additional time and support to those who have learned? Who will conduct the enrich-ments? When?
Proactive RTI Planning Form
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RTI
Forma)veAssessmentDifferen)a)on
WhatIstheConnecHon?
SessionLearningTargets
§ Wecaniden)fytheessen)alcontentstandardsstudentsmustlearninmathema)cs.
§ Wecaniden)fyhighcogni)vetaskstouseasforma)vefeedbackduringinstruc)on.
§ Wecandesignwaystoremediateandintervenewhenstudentsstruggletolearnmathema)cs.
Star)ng…Ge=ngThere…GotIt!
Star)ng…Ge=ngThere…GotIt!
Star)ng…Ge=ngThere…GotIt!
Thankyou!WhatquesHonsare
sHlllingering?
Whatisanextstepforyour
collaboraHveteamrelatedtoprioritystandardsandateamresponsetostudentlearning?
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